A symmetry operation obviously must not change the length of a vector or the angle between vectors. There are several concepts from the theory of metric spaces which we need to summarize. Example 2: a tensor of rank 2 of type (1-covariant, 1-contravariant) acting on 3 Tensors of rank 2 acting on a 3-dimensional space would be represented by a 3 x 3 matrix with 9 = 3 2 Deﬁnition:Ametric g is a (0,2) tensor ﬁeld that is: • Symmetric: g(X,Y)=g(Y,X). This is the second volume of a two-volume work on vectors and tensors. The above tensor T is a 1-covariant, 1-contravariant object, or a rank 2 tensor of type (1, 1) on 2 . Some Basic Index Gymnastics 13 IX. The resulting tensors may, however, prescribe abrupt size variations that Vectors and tensors in curved space time Asaf Pe’er1 May 20, 2015 This part of the course is based on Refs. is the metric tensor and summation over and is implied. This general form of the metric tensor is often denoted gμν. METRIC TENSOR 3 ds02 = ds2 (9) g0 ijdx 0idx0j = g0 ij @x0i @xk dxk @x0j @xl dxl (10) = g0 ij @x0i @xk @x0j @xl dxkdxl (11) = g kldxkdxl (12) The ﬁrst line results from the transformation of the dxiand the last line results from the invariance of ds2.Comparing the last two lines, we have Surface Covariant Derivatives 416 Section 57. When no so-lution is yet available, metrics based on the computational domain geometry can be used instead [4]. ijvi: It is said that “the metric tensor ascends (or descends) the indices”. 1 Tensor Analysis and Curvilinear Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: May 19, 2016 Maple code is available upon request. An open question regarding curvature tensors. In tensor analysis the metric tensor is denoted as g i,j and its inverse is denoted as g i,j. 2. covariant or contravariant, as the metric tensor facilitates the transformation between the di erent forms; hence making the description objective. 1. (The metric tensor will be expanded upon in the derivation of the Einstein Field Equations [Section 3]) A more in depth discussion of this topic can be found in [5]. 1Examples of tensors the reader is already familiar with include scalars (rank 0 tensors) and vectors (rank 1 tensors). That tensor, the one that "provides the metric" for a given coordinate system in the space of interest, is called the metric tensor, and is represented by the lower-case letter g. Definition Three different definitions could be given for metric, depending of the level - see Gravitation (Misner, Thorne and Wheeler), three levels of differential geometry p.199) Since G=M T M, 2.12 Kronekar delta and invariance of tensor equations we saw that the basis vectors transform as eb = ∂xa/∂xbe a. Understanding the role of the metric in linking the various forms of tensors1 and, more importantly, in diﬀerentiating tensors is the basis of tensor calculus, and the subject of this primer. Dual Vectors 11 VIII. Metric tensor Taking determinants, we nd detg0 = (detA) 2 (detg ) : (16.14) Thus q jdetg0 = A 1 q ; (16.15) and so dV0= dV: This is called the metric volume form and written as dV = p jgjdx1 ^^ dxn (16.16) in a chart. It does, indeed, provide this service but it is not its initial purpose. Since the matrix inverse is unique (basic fact from I feel the way I'm editing videos is really inefficient. 12|Tensors 2 the tensor is the function I.I didn’t refer to \the function (!~)" as you commonly see.The reason is that I(!~), which equals L~, is a vector, not a tensor.It is the output of the function Iafter the independent variable!~has been fed into it.For an analogy, retreat to the case of a real valued function new metric related the quantum geometry, ds̃2 = g AB dx A dxB, (8) where gAB = g ⊗ g . 4. His famous theorem, known to every student, is the basis for a remarkable thread of geometry that leads directly to Einstein’s3 Theory of Relativity. useful insight into metric tensors Afterwards, I asked what the diﬀerence betw een an outer product and a tensor product is, and wrote on the board something that lo oked like high-sc hool linear This latter notation suggest that the inverse has something to do with contravariance. Now consider G-1 X. immediately apparent from the components of the metric tensor which ones will allow coordinate transformations to get us to the unit matrix. Section 55. Example 6.16 is the tensor product of the ﬁlter {1/4,1/2,1/4} with itself. METRIC TENSOR: INVERSE AND RAISING & LOWERING INDICES 2 On line 2 we used @x0j @xb @xl @x0j = l b and on line 4 we used g alg lm= a m. Thus gijis a rank-2 contravariant tensor, and is the inverse of g ijwhich is a rank-2 covariant tensor. Observe that g= g ijdxidxj = 2 Xn i=1 g ii(dxi)2 +2 X 1≤i

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